de Broglie waves

De Broglie's move was an analogy, not a derivation from scratch. The photoelectric and Compton effects had already forced physicists to accept that a photon of wavelength $\lambda$ carries momentum $p = h/\lambda$. De Broglie simply inverted this relation and asserted it holds for any particle carrying momentum, light or matter alike:

where: $\lambda$ is the de Broglie wavelength, $h = 6.63\times10^{-34}\ \text{J}\,\cdot\,\text{s}$ is Planck's constant, and $p$ is the particle's momentum. For an everyday, non-relativistic particle of mass $m$ moving at speed $v$, momentum is simply $p = mv$, so the wavelength becomes

Often a particle's momentum is not given directly but built up from an accelerating potential difference. If a charge $e$ is accelerated from rest through a potential difference $V$, it gains kinetic energy $K = eV$, and since kinetic energy and momentum are related by $K = p^2/(2m)$ for non-relativistic motion, the momentum works out to $p = \sqrt{2mK} = \sqrt{2meV}$. Substituting gives the form used constantly in this chapter:

where: $m$ is the particle's mass, $e$ its charge, and $V$ the accelerating potential difference. Two ideas make this formula more than an algebraic curiosity. First, wave-particle duality is universal — the same electron, neutron, or football has both a well-defined momentum (its particle aspect) and an associated wavelength (its wave aspect); nothing singles out electrons as special. Second, the wave aspect is only observable when $\lambda$ is comparable to a relevant length scale, such as a slit width or a crystal's atomic spacing. Because $h$ is minuscule, macroscopic objects have wavelengths far too small to ever produce a detectable diffraction pattern, which is precisely why classical mechanics describes cars and cricket balls perfectly well. Electrons accelerated through tens to hundreds of volts, however, land in the angstrom range — the same scale as atomic lattice spacings — so their wave nature shows up directly as diffraction. The Davisson–Germer experiment (1927) supplied the proof. A beam of electrons, accelerated through a known potential, was scattered off the regularly spaced atomic planes of a nickel crystal, and the scattered intensity showed sharp maxima at specific angles — a Bragg-like diffraction pattern, exactly as X-rays diffract off a crystal. The wavelength extracted from those angles matched de Broglie's prediction $\lambda = h/p$ to good accuracy, giving direct experimental proof that matter has wave character.

An electron is accelerated from rest through a potential difference $V = 100\ \text{V}$. Take $m_e = 9.11\times10^{-31}\ \text{kg}$, $e = 1.6\times10^{-19}\ \text{C}$, and $h = 6.63\times10^{-34}\ \text{J}\,\cdot\,\text{s}$. Find its de Broglie wavelength, and comment on whether it is likely to diffract off a crystal.

Step 1. Kinetic energy gained: $K = eV = (1.6\times10^{-19})(100) = 1.6\times10^{-17}\ \text{J}$.

Step 2. Momentum from kinetic energy: $p = \sqrt{2m_eK} = \sqrt{2(9.11\times10^{-31})(1.6\times10^{-17})} = \sqrt{2.9152\times10^{-47}} = 5.40\times10^{-24}\ \text{kg}\,\cdot\,\text{m/s}$.

Step 3. de Broglie wavelength: $\lambda = h/p = (6.63\times10^{-34})/(5.40\times10^{-24}) = 1.228\times10^{-10}\ \text{m} = 1.228\ \text{Å}$.

Step 4. Compare to a crystal's atomic spacing, which is typically a few angstroms. The wavelength is $1.228\ \text{Å}$, squarely in the same range as the spacing between atomic planes — exactly why electrons accelerated through a hundred or so volts produce measurable diffraction in a Davisson–Germer-type setup, while a macroscopic object of the same kinetic energy would not.

Competitive stretch

A cricket ball of mass $0.16\ \text{kg}$ moves at $v = 30\ \text{m/s}$. Its momentum is $p = mv = 4.8\ \text{kg}\,\cdot\,\text{m/s}$, giving $\lambda = h/p = (6.63\times10^{-34})/4.8 = 1.38\times10^{-34}\ \text{m}$ — about twenty-four orders of magnitude smaller than the size of a proton, and utterly undetectable by any apparatus. This is why the cricket ball obeys ordinary Newtonian mechanics to extraordinary precision, while the $100\ \text{V}$ electron above must be treated as a wave.

• Treating $\lambda = h/p$ as applying only to electrons — it applies to any particle with momentum, including neutrons, atoms, and macroscopic bodies; the wave nature is simply unobservable when $\lambda$ is too small compared to relevant length scales.
• Using $p = \sqrt{2mK}$ without checking that the particle is non-relativistic; this simple relation assumes $v \ll c$ and needs modification at relativistic speeds, which lies outside this course.
• Confusing the de Broglie wavelength with the wavelength of light a particle might emit or absorb — the de Broglie wavelength is a property of the particle's own motion, entirely unrelated to any photon.
• Concluding from Davisson–Germer that electrons are "really" waves and not particles — the correct reading is wave-particle duality: the same electron shows wave-like diffraction in this experiment and particle-like localized detection in others.

Ask the tutor: want to see how the de Broglie wavelength changes if the electron's accelerating voltage is doubled, or how the Davisson–Germer angle relates to crystal spacing through Bragg's law?